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Multiplication of Matrices

If A is a s...

Question

If $A$ is a square matrix such that $A^{2} = A$ , then write the value of $7 A (I + A)^{3}$ , where $I$ is an identity matrix

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Solution

$7 A {(I + A)}^{3} = 7 A (I^{3} + A^{3} + I A + A I)$
$= 7 A (I + A^{2} \times A + 2 I A)$
$= 7 A (I + A \times A + 2 I A) [∵ A^{2} = A]$
$= 7 A (I + A^{2} + 2 I A)$
$= 7 A (I + A + 2 I A)$
$= 7 A I + 7 A \times A + 14 A I A$
$= 7 A + 7 A + 14 I (A A)$
$= 14 A + 14 I A$
$= 14 A + 14 A$
$= 28 A$

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